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

h smoothing length

𝑊

xi

• Function approximation

h

• Approximation of the Dirac Delta distribution:

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.

W(x-xi,h) kernel function

Smoothed Particle Hydrodynamics (SPH

• Properties of the kernel function

Unity

Compact support

Dirac Delta property

𝑊

𝑥𝑖

h

• Approximation of first order derivative

• Discrete approximation for derivatives of order n

(1)

(2)

Here the first term is neglected due to the shape of

Integrals are discretized and replaced by summation

Smoothed Particle Hydrodynamics (SPH)

subdomain of the particle in

Smoothed Particle Hydrodynamics (SPH

PROBLEM !!!

What about particles near to the boundary domain?

• When the smoothing length exceeds the boundary of the domain, the smoothing function is not completely developed

𝑊

hΩ

Smoothing function not completely developed in the domain

• In these cases Equation (2) does not hold and thus, derivative approximations are inaccurate

Reproducing Kernel Particle Method (RKPM)

such that

• Polynomial corrective functions are introduced

• 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

Here the property of unity is not necessary!

• Function approximation is used in the first derivative approximation. This may lead to error propagation

• Function approximation: the Taylor Series is expanded up to the zero-th order term

• First derivative approximation: the Taylor Series is expanded up to the first order term

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

• Coefficients 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 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 centered • Evaluate u(xj)-u(xi) in a set of supporting nodes .

• Collect these evaluations in a vector where is the number of the nodes supporting xi

• Evaluate five projection functions in the same supporting nodes xj

• Collect the evaluations of Wαi in five vectors

where

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 subjected to internal forces , constrained displacements on the Dirichlet boundary and tractions on the Neumann boundary .

• Dynamic equilibrium equations

Infinitely extended plate under traction

Ω

Stress component 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

Geometry and boundary conditions for a bar under impulsive traction

Convergence diagram of the error in stress for the original and novel formulation

3D problem: parallelepiped with spherical hole under traction

Stress component on the simmetry plane using MFPM (left) and using an overkilled Finite Element discretization (right)

Stress component on the simmetry plane using MFPM (left) and using an overkilled Finite Element discretization (right)

• The parallelepiped is under traction in the direction.

• , , are symmetry planes

• 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

Geometry of a parallelepiped with spherical bore with its simmetry planes

Dynamics: 2d bar under impulsive load

Geometry and boundary conditions for a bar under impulsive traction

Stress component in three time instant. The stress wave propagation can bee appreciated

Convergence diagram of the error in stress for the original and novel formulation

• 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 density material acceleration stress tensor body loads

pressure elastic shear modulus displacement (for elastic incompressible solids) velocity (for Newtonian fluids)

• Equilibrium equation

𝝈=−𝑝 𝑰+𝜇 (𝛻𝒖+𝛻𝒖𝑇 )

𝜌𝒂=−𝛻𝑝+𝜇Δ𝒖+𝒃

• Costrain equation

• Costitutive relation

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

• 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.

Simplified Pressure Poisson Equation(SPPE)

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.

• 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 is not neglected, and then the constrain equation is

In this case, according to Sani et al. (2006), no boundary conditions are required for the constrain equation, also at the discrete level.

• Boundary condition for the constrain equation

Consistent Pressure Poisson Equation – Formulation S2

Consistent Pressure Poisson Equation (CPPE)

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 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 . 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

• 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 S2 and MFPM

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

• 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 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)

• 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 is the diagonal matrix of the weights. They are identified accordingly with the particular discretization technique.

• 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.

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:

• LSRM has been applied on a square in the domain with the following exact solution

• Dirichlet boundary conditions have been imposed on the whole boundary

• Velocities and pressure have been discretized similarly.

• No pressure instability is obtained, differently from other collocation methods

Pressure field obtained using 231361 collocation points and 58081 control points

Application

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

Example of fully random distribution of collocation points (blue circles) and regular distribution of control nodes (green squares)

Convergence diagram of the error for three different randomly distributed collocation points

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 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 and

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.

• is the Reynolds number, that is defined as the ratio between the inertial and the viscous forces. It is computed as

where and are reference velocity and length respectively

Applications

𝑅𝑒=1000 𝑅𝑒=1000𝑅𝑒=400 𝑅𝑒=400

Lid driven cavity at : horizontal velocity profile (at ) compared with the reference solution by Ghia et al. (1982) (left); corresponding streamlines (right)

Lid driven cavity at : horizontal velocity profile (at ) 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.