space time FEM

8/12/2019 space time FEM

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Muhammad Sadiq Sarfaraz - Institut fr Statik

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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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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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

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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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Governing Equation:

Boundary conditions:

:solution variable:convection coefficient:diffusion coefficient

:source term

:independent variable


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Weak form:

where is the appropriate weighting function

Discretization of domain shown with isolated element


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Ansatz functions for solution variable ,independent

variable and weighting function for element interms of local coordinates variable

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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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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


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instability in solution

observed for


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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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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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Weighting functions for Upwind Type Finite Elements

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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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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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Stabilized Formulations

Modified Weak form:

where

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Variants of Stabilized Formulations based on

1. Galerkin Least squares (GLS) approach

2. Streamline Upwind Petrov Galerkin (SUPG) approach

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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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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

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Governing Equation:

Boundary conditions:

Weak Form:


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Nonlinearity: the convection coefficient is solution variable


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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


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1


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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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Exact solution for Burgers Equation with defined on

domain

stationary solution considered ( let )

Considering 20 elements with uniform spacing



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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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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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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 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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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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Definition of derivatives in terms of local coordinates for

the element

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Jacobian of transformation between local and global variables



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Stabilization term

For stabilization Galerkin Least Square formulation is used(Stabilized Finite element formulation)


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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



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with GLS stabilizationwithout stabilization

instability

Results for



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Application to Burgers Equation

Weak form for a Space-time slab

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Stabilization term for the element (GLS Stabilization)

Similarly stabilization parameter

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is a variable itself since it depends on solution variable



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with GLS stabilizationwithout stabilization

instability

Instability

damped

significantly

Space-time FEM for 1D Euler Equations


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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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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



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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)



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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)


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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



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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


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Time discontinuous Galerkin approach

solution discontinuous at nodes



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Weak form for a discrete element

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Ansatz function for solution variables, weighting functions and independent

variable in terms of local coordinates

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Final matrix form

unknown

known



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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



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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



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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



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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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Muhammad Sadiq Sarfaraz Institut fr Statik

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

space time FEM

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