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Muhammad Sadiq Sarfaraz
CSE Masters Student
Institut fr Statik
Technische Universitt Braunschweig
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Space-time Finite Element Method for OneDimensional Piston Cylinder System
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Objectives of Student Project
To demonstrate different approaches for Fluid Structure Interaction(FSI) problems using simplified model (based on ODEs) for Piston
cylinder system.
Implementation of Finite Element Method (FEM) for fluid problem inrelation with its application to Fluid part for Piston cylinder system
with increased complexity (1D Euler Equations)
Finally to develop a monolithic (coupled) framework for the
complete system using Space-time FEM
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1D Piston Cylinder System
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Demonstration of different approaches for FSI
problems using simplified model for Piston
cylinder system.
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Governing Equations
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: mass of piston
: spring stiffness: area of piston
: pressure
: displacement
: acceleration
: volume of cylinder: ratio of specific heats
spring mass system Isentropic gas law
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Reformulating variables for spring-mass system
Using the fact that two arbitrary states of gas can be
related through isentropic gas law and that the volumecan be expressed as a function of displacement the
pressure is given by:
Governing Equations
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( displacement )
( velocity )
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Governing Equations for Coupled Approach
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Differential Equations for
Obtained by differentiating
w.r.t time
Equations integrated in time as a coupled system
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Governing Equations for Elimination Approach
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Differential Equations for only. Pressure eliminated
from second equation.
Pressure is computed after temporal evolution of
is obtained from above equations
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Differential Equations for are same as considered
in Coupled approach, considering constant.
For a give time step proccess1 4 is carried out
repeatedly until
Governing Equations for Partitioned Approach
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1
2
3
4
1
Variable exchange
Step 1 for next time
step
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Results from Applied FSI Approaches
Input parameters
Constant parameters for Spring mass system
Constant parameters for Gas cylinder system
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Results from Applied FSI Approaches
Input parameters
Initial Conditions
Solver for time Integration: Matlab ode23t
applies Trapezoidal rulefor integration.
Symplecticin nature i.e. preserves geometrical structure of
solution.
Trapezoidal rule is A-stable and second order accurate
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Results from Applied FSI Approaches
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Displacement x(t)
For partitioned approach amplitude decreases in time
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Phase plot x(m) vs v(m/s)
For partitioned approach
locus of points
spirals inwardsi.e. demonstrating
dissipative behavior
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Isentropic law
For partitioned approach
locus of points
shrinks along the
hyperbolic curve
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Finite Element Formulation for Fluid problem
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Finite Element Formulation for Fluid problem
Objective: Implementation of FE formulation for gas in
cylinder governed by 1D Euler Equations
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InvestigateInstability
Proposal ofStabilizedschemes
FEM for
ConvectionDiffusionEquation
PicardScheme
NewtonScheme
FEM forNonlinear
ConvectionDiffusion
Equation AdvectionEquation
BurgersEquation
Space-time FEM
forTransientproblems
1
3
2
Follwing steps are applied:
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FEM for Convection Diffusion Equation
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Investigate Instability
Proposal of Stabilizedschemes
FEM for ConvectionDiffusion Equation 1
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FEM for Convection Diffusion Equation
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Governing Equation:
Boundary conditions:
:solution variable:convection coefficient:diffusion coefficient
:source term
:independent variable
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FEM for Convection Diffusion Equation
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Weak form:
where is the appropriate weighting function
Discretization of domain shown with isolated element
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FEM for Convection Diffusion Equation
Ansatz functions for solution variable ,independent
variable and weighting function for element interms of local coordinates variable
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FEM for Convection Diffusion Equation
Definition of derivatives in terms of local coordinates
where
: jacobi matrix defines transformation of derivatives
between local and global variables.
For given linear ansatz functions:
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FEM for Convection Diffusion Equation
The resulting element matrices and Load vector
Convection Matrix:
Diffusion Matrix:
Load Vector (assuming constant source term):
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Application to model problem:
Considering 10 uniform element giving
and is computed using Peclet number , which
determines the behavior of solution
Convection dominated
Diffusion dominated
Analytical solution for and
FEM for Convection Diffusion Equation
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FEM for Convection Diffusion Equation
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instability in solution
observed for
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FEM for Convection Diffusion Equation
Cause of instability for Convection dominated case ( )
Standard Galerkin approach does not account for the direction
of flow governed by convection coefficient
Numerically: the amount of diffusion introduced by Galerkinapproach is less than the required amount, to get a stable
solution.
Approximation of convection term is same as taking its central
difference approximation in Finite difference method which isunstable.
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same as unstable
central difference
approximation
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FEM for Convection Diffusion Equation
Alternate FEM Formulations studied:
1. Upwind Type Finite Elements.
2. Streamline Upwind (SU) Type Finite Elements.
3. Stabilized Formulations.
In (1) and (2) the ansatz function is different for weighting
functions as compared to element solution approximation. (3) is based on addition of stabilization term in standard
formulation .
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FEM for Convection Diffusion Equation
Weighting functions for Upwind Type Finite Elements
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FEM for Convection Diffusion Equation
Weighting functions for Streamline Upwind (SU) Type
Finite Elements
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FEM for Convection Diffusion Equation (SU type FEM)
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Results using
optimal value
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FEM for Convection Diffusion Equation
Remarks on
is vital for accurate solution , since it scales the additional
diffusion required for stable solution.
Optimal value for yields exact solution at nodes. For more
general cases(e.g. unknown exact solution ,variable source
term) it is not available explicitly.
Remarks on Upwind and SU type FEM schemes:
Upwind schemes are more diffusive as compared to SU
schemes.
However SU schemes result in non residual formulation, (alsocalled Inco ns istent SU ) hence do not perform well for general
cases (see results for Stabilized schemes)
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FEM for Convection Diffusion Equation
Stabilized Formulations
Modified Weak form:
where
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FEM for Convection Diffusion Equation
Variants of Stabilized Formulations based on
1. Galerkin Least squares (GLS) approach
2. Streamline Upwind Petrov Galerkin (SUPG) approach
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FEM for Convection Diffusion Equation
Stabilization parameter
For given problem ( ):
Expression for derived for the given problem using exact
solution and numerical scheme.
Such analytical expression may not available for more general
cases (unknown exact solution or complex source term )
For system of differential equations assumes the form ofmatrix
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FEM for Convection Diffusion Equation
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Note: For SUPG the same is used ,which was derived for
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FEM for Nonlinear Convection Diffusion Equation
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Picard Scheme Newton Scheme
FEM for NonlinearConvection DiffusionEquation
2
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Governing Equation:
Boundary conditions:
Weak Form:
FEM for Nonlinear Convection Diffusion Equation
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Nonlinearity: the convection coefficient is solution variable
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FEM for Nonlinear Convection Diffusion Equation
Element nonlinear convection matrix from FEM discretization:
Nonlinearity: solution variable appearing in convection matrix
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Solution strategies for Nonlinear System of Equations
For PICARD iteration scheme the assembly procedure is same
as for linear case i.e stiffness matrices for elements are
assembled to get their nonlinear Global counterpart
Solve for iteratively until
: successive iteration levels
: specified vector norm to measure the error
:specified tolerance to end the iterative procedure
FEM for Nonlinear Convection Diffusion Equation
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1
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FEM for Nonlinear Convection Diffusion Equation
Solution strategies for Nonlinear System of Equations
For NEWTON scheme the Global residual vector is
expanded around using Taylor series and set to zero
neglecting higher order terms
where
Solution update:
Procedure continues until
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2
(Tangent stiffness matrix)
FEM f N li C i Diff i E i
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FEM for Nonlinear Convection Diffusion Equation
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Exact solution for Burgers Equation with defined on
domain
stationary solution considered ( let )
Considering 20 elements with uniform spacing
FEM for Nonlinear Convection Diffusion Equation
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FEM for Nonlinear Convection Diffusion Equation
Results
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Picard Newton
Convergence
comparisonNewton method converges
faster
Quadratic or higher degree
ansatz functions required for
further error reduction
S ti FEM f T i t bl
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Space-time FEM for Transient problems
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Advection Equation
Burgers Equation
Space-time FEM
for Transient problems3
S ti FEM f T i t bl
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Space-time FEM for Transient problems
Considering 1D transient Convection diffusion Equation
with certain initial and boundary conditions
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S ti FEM f T i t bl
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Space-time FEM for Transient problems
Discretization of Domain in Space-time FEM
In Space-time FEM, the time dimension is treated as a spatialdimension.
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Discretization of domain with isolated element
S ti FEM f T i t bl
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Space-time FEM for Transient problems
Space-time discontinuous Galerkin formulation
Considers elements to be discontinuous in time dimension
Continuity in time is weakly enforced as shown in weak form for
a discrete element
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Interface of elements at time level
enforces inter-element continuity in time
S ti FEM f T i t bl
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Space-time FEM for Transient problems
Element ansatz functions for weighting function , solution
approximation and independent variables , in terms of
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Space time FEM for Transient problems
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Space-time FEM for Transient problems
Definition of derivatives in terms of local coordinates for
the element
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Jacobian of transformation between local and global variables
Space time FEM for Transient problems
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Space-time FEM for Transient problems
Stabilization term
For stabilization Galerkin Least Square formulation is used(Stabilized Finite element formulation)
Stabilization parameter
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Space time FEM for Transient problems
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Space-time FEM for Transient problems
Application to Linear Advection Equation
Sdd
100 elements considered for spatial domain
Time evolution till seconds. Spacing of element in
time is based on Courant number
Result are shown for
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Time evolution of a unit step profile
Space-time FEM for Transient problems
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Space-time FEM for Transient problems
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with GLS stabilizationwithout stabilization
instability
Results for
Space time FEM for Transient problems
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Space-time FEM for Transient problems
Application to Burgers Equation
Weak form for a Space-time slab
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Space time FEM for Transient problems
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Space-time FEM for Transient problems
Stabilization term for the element (GLS Stabilization)
Similarly stabilization parameter
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is a variable itself since it depends on solution variable
Space-time FEM for Transient problems
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Space time FEM for Transient problems
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with GLS stabilizationwithout stabilization
instability
Instability
damped
significantly
Space-time FEM for 1D Euler Equations
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Space-time FEM for 1D Euler Equations
Now implementing FEM for 1D Euler Equations since we already
have
Investigated stability issues regarding FEM implementation for
Fluid problems and proposed remedial measures
Demonstrated solution schemes for nonlinear system of
equations arising from FEM application to nonlinear problems
FEM method for transient problems: application to both, linear
and nonlinear model problems
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Space-time FEM for 1D Euler Equations
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Space-time FEM for 1D Euler Equations
1D Euler Equations in Conservative form
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Solution variable vector (mass, momentum, energy)
Flux jacobian matrix
Total Energy per unit mass
Total Enthalpy per unit mass
Ideal gas law
Ratio of specific heat values
Flux vector
Space-time FEM for 1D Euler Equations
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Space-time FEM for 1D Euler Equations
Weak form for a Space-time Slab (element)
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Stabilization term (GLS formulation)
Inter-element continuity enforced in time(Time discontinuous Galerkin formulation)
Space-time FEM for 1D Euler Equations
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Space-time FEM for 1D Euler Equations
Definition of ansatz functions for weighting function, solution variables
and independent variables remain the same as discussed for advection
and Burgers Equation
The definition of jacobian also remains unaltered
Only difference is in the structure of resulting matrics since we are
dealing with system of Equations (mass, momentum and energy)
Stabilization parameter
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Space-time FEM for 1D Euler Equations
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Space-time FEM for 1D Euler Equations
Application to Shock tube problem
Domain:
Time evolution for:
Element spacing:
Constant parameter and gas constant assume the values that of Air
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Initial state at
Space-time FEM for 1D Euler Equations
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Space time FEM for 1D Euler Equations
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Total Energy Density
Velocity Pressure
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Space-time FEM for Piston cylinder system
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Space-time FEM for Piston cylinder system
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Space time FEM for Piston cylinder system
Objective :To develop a monolithic framework for the given problem using
Space-time FEM
Steps required:
1. FEM implementation for Structural part. (Spring mass system)
2. Space-time FEM formulation for Euler Equation. (modification required
to incorporate deforming domain)
3. Coupling between the Structural and Fluid part.
4. Solution procedure and Mesh update.
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FEM for Spring mass system
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The governing equations for are recast in form consistent with FEM
Formulation
FEM for Spring mass system
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Time discontinuous Galerkin approach
solution discontinuous at nodes
FEM for Spring mass system
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FEM for Spring mass system
Weak form for a discrete element
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FEM for Spring mass system
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FEM for Spring mass system
Ansatz function for solution variables, weighting functions and independent
variable in terms of local coordinates
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Final matrix form
unknown
known
FEM for Spring mass system
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Results from some test cases
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critically
damped
system
damped
System with
force term
System with no force
and damping term
Space-time FEM for 1D Euler Equations
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Using primitive (non conservative) form of Euler Equations for
which:
The standard matrix form for Equations remains same as for
conservative case.
Primitive form makes the coupling procedure simpler.
Space -time discontinuous Galerkin approach is used.
p q
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Primitive
variables:
1.Density
2.Velocity
3. Pressure
is not the Flux
jacobian in primitive
form
Space-time FEM for 1D Euler Equations
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p q
How to account for the deforming domain ?
Answer: Space-time FEM takes care of deforming domainautomatically through the definition of jacobian matrix
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Fluid domain deformation during time step Deformed element
Space-time FEM for 1D Euler Equations
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p q
Definition of Jacobian for the element in deforming domain
Hence the mapping from the global to local domain is handled by
Jacobian
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Unlike the non deforming domain
these elements of are variables
Coupling between Structural and Fluid problems
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p g p
For Monolithic(Coupled) FSI approach the Global matrices are
assembled as one single system
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Structure of Global matrix at interface node 1
Solution procedure and Mesh update
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p p
Steps followed:
For given time step the spatial nodes
are considered explicitly known.
The Global nonlinear system is solved
using Picard scheme.
The nodal positions for Fluid domain are updated
using displacement at node 1 (from structure)
employing linear interpolation.
The steps above are performed repeatedly
until desired convergence is achieved