The Modified Finite Particle Method
for incompressible solids and fluids
D. Asprone, F. Auricchio, A. Montanino, A. Reali
Lille, January 22th, 2014
Outline
• Review of the SPH derived methods
• The Modified Finite Particle Method (MFPM)
• Application to elasticity
• Incompressible elasticity equations
• Discretization with collocation methods: numerical difficulties
• Alternative formulations of the incompressile elasticity equations
• Applications with MFPM
• MFPM in the framework of the Least Square Residual Method
• Solution of non linear problems
Smoothed Particle Hydrodynamics (SPH)
• Approximation starting point: Dirac Delta equivalence
Dirac Delta distribution, centered in xi
• Approximation of the Dirac Delta distribution:
𝑊
• Function approximation
W(x-xi,h)
kernel function
h
smoothing length
xi
ℎ
Lucy, L. (1977). A numerical approach to the testing of the fission hypothesis. The astronomical journal
82, 1013–1024.
Gingold, R. and J. Monaghan (1977). Smoothed Particle Hydrodynamics: theory and application to
non spherical stars. Monthly Notices of the Royal Astronomical Society 181, 375–389.
Smoothed Particle Hydrodynamics (SPH
• Properties of the kernel function
Unity
Compact support
Dirac Delta property
𝑊
𝑥𝑖
ℎ
Smoothed Particle Hydrodynamics (SPH)
• Approximation of first order derivative
(1)
Here the first term is neglected due to the shape of 𝑊(𝑥 − 𝑥𝑖 , ℎ)
• Discrete approximation for derivatives of order n
(2)
Integrals are discretized and replaced by summation
subdomain of the particle in 𝑥𝑖
Smoothed Particle Hydrodynamics (SPH
PROBLEM !!!
What about particles near to the boundary domain?
𝑊
Smoothing
completely
domain Ω
function 𝑊
developed in
not
the
Ω
ℎ
• When the smoothing length exceeds the boundary of the domain, the
smoothing function is not completely developed
• In these cases Equation (2) does not hold and thus, derivative approximations
are inaccurate
Reproducing Kernel Particle Method (RKPM)
• Polynomial corrective functions 𝐶𝑖 (𝑥) are introduced
such that
• In this way, the consistency of of the method is restored also on the boundary.
• The order of the polynomial Ci(x) determines the order of the polynomial that
can be exactly reproduced by RKPM
• With the RKPM no improvement is given to the accuracy of derivative
approximation
Liu, W. K., S. Jun, S. Li, J. Adee, and T. Belytschko (1995). Reproducing Kernel Particle Methods for
Structural Dynamics. International Journal for Numerical Methods in Engineering 38, 1655–1680.
Liu, W. K., S. Jun, and Y. F. Zhang (1995). Reproducing Kernel Particle Methods. International Journal for
Numerical Methods in Fluids 20, 1081–1106.
Corrective Smoothed Particle Method (CSPM)
• Starting point: Taylor series expansion
• Function approximation: the Taylor Series is expanded up to the zero-th
order term
Here the property of unity is not necessary!
• First derivative approximation: the Taylor Series is expanded up to the first
order term
• Function approximation is used in the first derivative approximation. This may
lead to error propagation
Chen, J. K., J. E. Beraun, and T. C. Carney (1999). A Corrective Smoothed Particle Method for boundary
value problems in heat conduction. International Journal for Numerical Methods in Engineering 46, 231–
252.
Chen, J. K., J. E. Beraun, and C. J. Jih (1999). An improvement for tensile instability in Smoothed Particle
Hydrodynamics. Computational Mechanics 23, 279–287.
Modified Smoothed Particle Method (MSPH)
• Function and derivatives are approximated simultaneously. The Tayor Series
is projected on the kernel function and its first derivative, obtaining
𝑖
• Coefficients 𝐴𝑗𝑙
are derived from the Taylor series:
• Second order accuracy is got in the interior of the domain
• First order accuracy is got at the boundary
Zhang, G. and R. Batra (2004). Modified smoothed particle hydrodynamics method and its application to
transient problems. Computational Mechanics 34, 137–146.
Modified Finite Particle Method (MFPM)
• Starting point: MSPH
• Consider the two dimensional Taylor series of a function 𝑢(𝒙) centered in a
point xi
• 5 unknown derivatives have to be approximated
• 5 projection functions
for α=1,…,5 are introduced
• No unity property is required
• No compact support property is required
• No Dirac Delta property is required
Modified Finite Particle Method (MFPM)
• Taylor series is projected on the projection functions
𝛼 = 1, … , 5
• In matrix form
Modified Finite Particle Method (MFPM)
• Integrals have to be discretized.
• Voronoi procedure is used to divide the domain in subdomains.
• Each particle
is assigned to a subdomain
• Integrals are replaced by summation
• Inverting this system, we get derivative approximations
PROBLEMS
• The Voronoi tessellation procedure is time consuming
• Numerical errors introduced when discretizing integrals
Novel Modified Finite Particle Method
• Consider the Taylor series expansion of 𝑢(𝒙) centered 𝒙𝑖
• Evaluate u(xj)-u(xi) in a set of supporting nodes 𝒙𝑗 .
• Collect these evaluations in a vector
where
is the number of the nodes supporting xi.
• Evaluate five projection functions
nodes xj
• Collect the evaluations of Wαi in five vectors
where
in the same supporting
Novel Modified Finite Particle Method
• Scalar product
𝛼 = 1, … , 5
• In matrix form
• Inverting the system,
derivative approximations
are obtained
ADVANTAGES
• No integral discretization is needed
• No Voronoi tessellation is required
Linear elasticity
• Consider a linear elastic body Ω subjected to internal forces 𝒃 = 𝒃(𝒙) ,
constrained displacements 𝒖 = 𝒖(𝒙) on the Dirichlet boundary Γ𝐷 and
tractions 𝒕 = 𝒕(𝒙) on the Neumann boundary Γ𝑁 .
• Dynamic equilibrium equations
Infinitely extended plate under traction
Ω
Geometry
and
conditions for a
impulsive traction
boundary
bar under
Convergence diagram of
the error in stress for the
original
and
novel
formulation
Stress component 𝜎𝑥𝑥
Obtained with MFPM
• Second order convergence of the error in both the original and novel
formulation
• The error costant is reduced in the novel formulation
3D problem: parallelepiped with spherical hole under traction
Stress component 𝜎𝑥𝑥 on the simmetry plane 𝑥𝑧 using
MFPM (left) and using an overkilled Finite Element
discretization (right)
Geometry of a parallelepiped with
spherical bore with its simmetry planes
• The parallelepiped is under
traction in the 𝑥 direction.
• 𝑥=0, 𝑦=0, 𝑧=0
symmetry planes
are
Stress component 𝜏𝑥𝑦 on the simmetry plane 𝑥𝑧 using
MFPM (left) and using an overkilled Finite Element
discretization (right)
• The novel formulation is easily extensible to 3D, while the original formulation
implies 3D Voronoi tessellation that is difficoult to program and extremely time
consuming
Dynamics: 2d bar under impulsive load
Geometry and boundary conditions for a
bar under impulsive traction
Convergence diagram of the error in
stress for the original and novel
formulation
Stress component 𝜎𝑥𝑥 in three time instant. The
stress wave propagation can bee appreciated
• Also in the case of dynamics, second order spatial accuracy is achieved both
for original and novel formulation.
References about MFPM in elasticity
Asprone, D., F. Auricchio, G.Manfredi, A. Prota, A. Reali, and G. Sangalli
(2010). ParticleMethods for a 1d Elastic Model Problem: Error Analysis and
Development of a Second-Order Accurate Formulation. Computational
Modeling in Engineering & Sciences 62, 1–21.
Asprone, D., F. Auricchio, and A. Reali (2011). Novel finite particle formulations
based on projection methodologies. International Journal for Numerical
Methods in Fluids 65, 1376–1388.
Asprone, D., F. Auricchio, and A. Reali (2014). Modified finite particle method:
applications to elasticity and plasticity problems. International Journal of
Computational Methods 11(01).
Asprone, D., F. Auricchio, A. Montanino, and A. Reali (2014). A Modified Finite
Particle Method: Multi-dimensional elasto-statics and dynamics. International
Journal for Numerical Methods in Engineering 99(01), 1-25.
Incompressibility
• Indefinite equilibrium equation: as for compressible bodies
𝜌𝒂 = 𝛻 ⋅ 𝝈 + 𝒃
𝜌
𝝈
density
stress tensor
𝒂
𝒃
material acceleration
body loads
• Costitutive relation
𝝈 = −𝑝𝑰 + 𝜇 (𝛻 𝒖 + 𝛻 𝒖𝑇 )
𝑝
𝒖
pressure
𝜇 elastic shear modulus
displacement (for elastic incompressible solids)
velocity (for Newtonian fluids)
• Equilibrium equation
𝜌 𝒂 = −𝛻𝑝 + 𝜇Δ𝒖 + 𝒃
• Costrain equation
𝛻⋅𝒖=0
Lagrangian VS Eulerian point of view
• Here we focus on elastic fluids, then
material acceleration.
Two points of view:
• Lagrangian point of view: the observer follows body particles during thier
motion.
• Eulerian point of view: the observer is in a fixed position during the fluid
motion
In this case
convective velocity
Stokes and Navier Stokes Equations
• Stokes equations
• Describing highly-viscous flows in an Eulerian point of view
• Describing all flows in Lagrangian point of view
• Navier-Stokes equations
• Describing all flows in an Eulerian point of view
• non linear equations
Stokes and Navier Stokes Equations
• Properties of the equations
Stokes and Navier-Stokes Equations
are parabolic problems with a
constrain equation.
For its solution, one initial condition is required on the whole domain;
two boundary conditions are required at each boundary
• Possible boundary conditions
•
•
Dirichlet boundary conditions (known velocity at the boundary)
Neumann boundary conditions (known stress)
• No boundary conditions are required for the constrain equation
Stokes and Navier Stokes Equations: discretization
• Adopting the same interpolation of velocity and pressure leads to a well
known instability of the pressure field.
• This instability has been first studied by Brezzi and Fortin (1991), which
estabilished a condition that has to be respected in order to avoid pressure
instability. This condition is known as LBB (Ladyzenskaia-Babushka-Brezzi)
condition or inf-sup condition
• In Finite Element Method, usually velocity is discretized using quadratic
interpolation, and pressure using linear interpolation.
• In the field of Finite Difference method, the
difficulty is overcome using staggered grids,
that is, velocity and pressure are discretized
in different nodes, and also the equilibrium
equations and incompressibility constrains
are collocated in different points.
Brezzi, F. and M. Fortin (1991). Mixed and hybrid finite element methods. Springer-Verlag New York, Inc.
Stokes and Navier Stokes Equations using MFPM
• Staggered grids cannot be used in meshless methods, where nodes could be
randomly distributed.
• Different formulations should be used in order to solve incompressibility
equations on non staggered grids.
• In the following we analyze five different formulations proposed in the
literature. Velocity and pressure are discretized using MFPM. We will restrict
to the stationary, linear case, where the difficulties connected with the inf-sup
condition are still evident.
Simplified Pressure Poisson Equation – Formulation S1
• The incompressibility constrain is replaced by a different condition obtained
applying the divergence operator to both sides of the equilibrium equation
that is
Simplified Pressure Poisson Equation
(SPPE)
• Boundary condition for the constrain equation
• According to Sani et al. (2006), at the discrete level, a boundary condition is
needed also for the constrain equation. The authors propose, as boundary
condition, the projection of the equilibrium equation on the outward
normal at the boundary
Sani, R. L., J. Shen, O. Pironneau, and P. Gresho (2006). Pressure boundary condition for the timedependent
incompressible navier–stokes equations. International Journal for Numerical Methods in Fluids 50(6), 673–682.
Consistent Pressure Poisson Equation – Formulation S2
• Also in this case, the incompressibility constrain is replaced by a different
condition obtained applying the divergence operator to both sides of the
equilibrium equation
• In this case, the term 𝛻 ⋅ Δ𝐮 is not neglected, and then the constrain equation
is
Consistent Pressure
Poisson Equation (CPPE)
• Boundary condition for the constrain equation
In this case, according to Sani et al. (2006), no boundary conditions are
required for the constrain equation, also at the discrete level.
Sani, R. L., J. Shen, O. Pironneau, and P. Gresho (2006). Pressure boundary condition for the timedependent
incompressible navier–stokes equations. International Journal for Numerical Methods in Fluids 50(6), 673–682.
Formulation S3
• In this case, similarly to formulation S1, the incompressibility constrain is
replaced by the Simplified Pressure Poisson Equation
• Boundary condition for the constrain equation
• In formulation S3, the boundary condition used for the constrain equation is
the original incompressibility equation
• Formulation S3 is then
in the domain
in the domain
on the Dirichlet boundary
on the Neumann boundary
on the whole boundary
Relaxed incompressibility constrain – Formulation S4
• In this formulation, the incompressiblity constrain is replaced by a relaxed
constrain
The Laplacian of the pressure «smooths» the pressure field, avoiding the
typical oscillations of the original formulation. The parameter 𝜀 has to be
properly set in order not to modify too much the problem.
• Boundary condition for the constrain equation
• In formulation S4, the boundary condition used for the constrain equation is
the original incompressibility equation, without the smoothing term
Gauge Method – Formulation S5
• The Gauge Method (Wang and Liu, 2000; E and Liu, 2003) is based on a
change of variables
• This change of variables permits to decouple pressure and velocities in the
equilibrium equation. The inf-sup condition, in this case, has not to be
respected.
Wang, C. and J.-G. Liu (2000). Convergence of Gauge Method for incompressible flow. Mathematics of
computation 69(232), 1385–1407
E, W. and J.-G. Liu (2003). Gauge Method for viscous incompressible flows. Communications in
Mathematical Sciences 1(2), 317–332.
Gauge Method – Formulation S5
• Boundary conditions
• Boundary conditions are modified according to the change of variables of the
Gauge Method
Dirichlet BC
• When the first condition is split in two parts, we obtain a condition for the
Gauge variable 𝜙. The final set of boundary conditions are thus
Application with MFPM
• Formulations S1, S2 and S3 do not impose directly the incompressibility
constrain, but a derived one. Here we verify if the incompressibility condition
is respected or not.
• We consider the well-known problem of the lid-driven cavity
• The normal velocity is zero on the whole boundary
• The tangential velocity is zero on the whole boundary,
except than on the upper side, where
Divergence of the velocity with formulations S1, S2 and S3
Divergence of the velocity obtained
with formulation S1 and MFPM
Divergence of the velocity obtained
with formulation S2 and MFPM
• From the figures, we see that the only
formulation in which the incompressibility
constrain is substantially respected is the
formulation S3. For this reason, formulations
S1 and S2 are not further investigated
Divergence of the velocity obtained
with formulation S3 and MFPM
Application with MFPM
• The order of convergence of MFPM applied on formulations S3, S4 and S5 is
computed on a Stokes problem with exact solution
with 𝑥, 𝑦 ∈ [−1,1]x[−1,1]
• Dirichlet boundary conditions are imposed on the whole boundary, and are
computed in accordance to the exact solution.
• From the figure, we see that all the
described formulation, implemented
with MFPM for the spatial discretization,
show second-order accuracy
Least Square Residual Method (LSRM)
• A different way for approaching incompressible elasticity problems is the
Least Square Residual Method (LSRM), used in combination with the
Radial Basis Functions by Chi et al. (2014)
• Given the linear system of equations (representing, for example, the
discretization of a partial differential equation)
if the number of equations is greater than the number of unknowns, the
system is overdetermined and 𝒖 is found by minimizing the squared error
that is,
• This system is a well posed, symmetric system of equations, and can be
solved using specific algorithms for symetric matrices
Chi, S.-W., J.-S. Chen, and H.-Y. Hu (2014). A weighted collocation on the strong form with mixed radial
basis approximations for incompressible linear elasticity. Computational Mechanics 53(2), 309–324
Weighted Least Square Residual Method (WLSRM)
• To apply LSRM to partial differential equations, the number of equations
must be greater than the number of unknows. This means that the number
of collocation points (the points where the discretized equations are written)
has to be higher than the number of nodal unknowns.
• It also should be considered that some equations are the discretization of
boundary conditions, and can have a different order of magnitude with respect
to the field equations.
• Some weights are thus included in the formulation, in order to restore the
same order of magnitude. The previous system of equations is then modified
in the form
where 𝑾 = diag[𝑤1 , 𝑤2 , … , 𝑤𝑁 ] is the diagonal matrix of the weights. They are
identified accordingly with the particular discretization technique.
Weighted Least Square Residual Method (WLSRM)
• For the Stokes problem, using Radial Basis Functions for the spatial
discretization, Chi et al. (2014) propose
where
are the weights associated to Dirichlet BC
are the weights associated to Neumann BC
are the weight associated to incompressibility constrain
is the second Lamé constant (viscosity of the fluid)
is the number of nodal unknowns
Weighted Least Square Residual Method and MFPM
• The Weighted Least Square Residual Method (WLSRM) can be used also
in combination with the Modified Finite Particle Method (MFPM).
• A little modification has to be made to the original formulation, since here
collocation points and nodal unknowns have to be decoupled (that was not
possible in the original formulation).
• In a 2D domain, for each node, the function approximation, as well as the
derivative approximations have to be determined through the system
are the collocation points
are the control nodes, that is, the nodes on which variables are evaluated
Weighted Least Square Residual Method and MFPM
• Collocation points are placed following the geometry of the problems. Control
nodes can be placed in any way, and so, also on a regular background grid
• The
weights
proposed
for
the
implementation of the Weighted LSRM,
using the MFPM for the spatial
discretizations, are:
Application
• LSRM has been applied on a square in the domain 𝑥, 𝑦 ∈ −1,1 x[−1,1] with
the following exact solution
• Dirichlet boundary conditions have been imposed on the whole boundary
• Velocities and pressure
discretized similarly.
have
been
• No pressure instability is obtained,
differently from other collocation methods
Pressure field obtained using 231361
collocation points and 58081 control
points
Application
• For the same problem, the obtained convergence diagram of the error
confirms the second order accuracy of the method
Convergence diagram of the error for
regular distributed collocation points
Convergence diagram of the error for three
different randomly distributed collocation
points
Example of fully random distribution
of collocation points (blue circles)
and regular distribution of control
nodes (green squares)
Application
• On the quarter of annulus in figure, a polynomial solution is given, and
accordingly, Dirichlet boundary conditions and internal body forces are
assigned.
Quarter of annulus
under body load:
horizontal velocity
field obtained with
MFPM and LSRM
Quarter of annulus under
body load: convergnce
diagram of the error
Quarter of annulus under
body load: vertical velocity
field obtained with MFPM
and LSRM
Non-linear problems
• Navier-Stokes Equations
• The solution can be obtained through linearization according to the Picard or
the Newton-Raphson algorithm. The final system is of the type
is the linearized stiffness matrix
is the increment of the unknown
is the residual of the equation
• If the number of collocation points is greater than the number of control
nodes, this system is rectangular, and thus, it has to be solved, similarly to the
linear case, using the LSRM approach. Also in this case, weights are required
to balance the error contributes.
Applications
• Using the Least Square Residual Method and the Modified Finite Particle
Method, we solve the problem of the lid-driven cavity and the problem of the
flow past a cylinder
• Lid-driven cavity
• The problem is a square cavity in the domain
−0.5,0.5 x[−1,0] with all sides clamped except the top
side, that has a tangential velocity different from zero.
• Our results are compared with those obtained in the
literature (Ghia et al, 1982) for 𝑅𝑒 = 400 and 𝑅𝑒 = 1000
• 𝑅𝑒 is the Reynolds number, that is defined as the ratio
between the inertial and the viscous forces. It is
computed as
where 𝑉 and 𝐿 are reference velocity and length
respectively
Ghia, U., K. N. Ghia, and C. Shin (1982). High-re solutions for incompressible flow using the navier-stokes
equations and a multigrid method. Journal of computational physics 48(3), 387–411.
Applications
Lid driven cavity at 𝑅𝑒 =
400 : horizontal velocity
profile (at 𝑥 = 0) compared
with the reference solution
by Ghia et al. (1982) (left);
corresponding streamlines
(right)
𝑅𝑒 = 400
𝑅𝑒 = 400
𝑅𝑒 = 1000
𝑅𝑒 = 1000
Lid driven cavity at 𝑅𝑒 =
1000 : horizontal velocity
profile (at 𝑥 = 0) compared
with the reference solution
by Ghia et al. (1982) (left);
corresponding streamlines
(right)
Applications
• Flow past a cylinder
• Here we solve the flow past a cylinder. No reference results are available in
the literature, but we can appreciate the smoothness of the solution.
Flow past a cylinder: gemometry and
boundary conditions
Flow past a cylinder: horizontal velocity solution
obtained with MFPM in combination with LSRM
Conclusion
• In this work, we apply the Modified Finite Particle Method to compressible
linear elasticity, incompressibile elasticity, fluid dynamics
• For linear elastic problems, in the tests where an analytical solution is
available, MFPM shows correct second-order accuracy
• When approaching incompressible elasticity, MFPM, similarly to other
numerical methods, suffers from spurious oscillations of the pressure, unless
the so-called inf-sup condition is respected, or rather, alternative formulations
are introduced
• In formulations where the incompressibility constrain is respected (in this
presentation, formulation S3-S5), the second-order accuracy of the method is
confirmed
• When MFPM is used in combination with a Weighted Residual Least-Square
Method (WLSRM), the solution of linear problems on complicated geometries
and/or randomly distributed set of nodes leads to the expected second-order
accuracy.
• Also the solution of the non-linear Navier-Stokes problem give accurate
solutions, using MFPM with the WLSRM approach.