Transcript The use of Spectral Methods in Fluid

Presented by: Mario Lee
Class: MATH 6645
Overview of the FSI problem
 Simple example: A flag fluttering in the breeze
Referenced Papers:
• Kollmannsberger, Stefan. ALE-type and fixed grid fluid-structure interaction
involving the p-version of the Finite Element Method. München, Techn. Univ.,
Diss., 2010.
• Ho, Lee Wing. A Legendre spectral element method for simulation of
incompressible unsteady viscous free-surface flows. Diss. Massachusetts
Institute of Technology, 1989.
Examples of Practical FSI Problems
Wind turbine analysis
Ship hull analysis
Aircraft aeroelastic analysis
Cardiovascular system analysis
Methods of Analyzing FSI Problems
 Monolithic approach
Treats the fluid and structure dynamics in the same mathematical framework
to form of a single system of equations for the entire problem
• Potentially achieves better accuracy but requires more development time
 Partitioned Approach
• Treats the fluid and the structure as two computational fields which can be
solved separately with their respective mesh discretization and numerical
algorithms
• An interface is used to communicate information between the two domains
•
Definition of Problem Domain
Definition of the Problem Domain
Ω𝑓 :- Fluid domain
Ω𝑠 :- Solid domain
Ω = Ω𝑓 ∪ Ω𝑠 :- Total domain
Γ𝑠 = Ω𝑓 ∩ Ω𝑠 :- Fluid-structure interface
Mechanics of the Fluid Domain
 Strong form: Navier -Stokes Equation
Incompressible case is considered
• Newtonian fluid
•
(1)
(2)
Definition of variables:
v – velocity of the fluid
𝜌 – density of the fluid
p – pressure of the fluid
σ −shear stress tensor
𝐼 − identity tensor
Lagrangian and Eulerian Coordinates
 Lagrangian reference frame
•
•
Reference frame tries to track the path and velocity of each individual particle
Disadvantage: Becomes intractable for problems with large deformations[3]
 Eulerian reference frame
•
•
Reference frame sets up fixed regions and monitors the behavior of the particles flowing in
and out of the region
Disadvantage: Requires fine meshes and as such, can be computationally burdensome[3]
 Arbitrary Lagrangian-Eulerian reference frame
•
•
•
Allows the mesh to move at an arbitrary rate, independent of the solution of the problem. [3]
The mesh is allowed to be distorted so that there is not need for re-meshing at each time
step[1]
Allows for optimization of the shape of the elements[1]
Weak Form of the Governing Equations
Strong Form
Weak Form
 Problem statement
• Find
1
2
• 𝑣 ∈ 𝐻0 [Ω𝑓 𝑡 ] and 𝑝 ∈ 𝐿 Ω𝑓 𝑡 ;
1
2
• ∀𝛿𝑣 ∈ 𝐻0 [Ω𝑓 𝑡 ] and ∀𝛿𝑞 ∈ 𝐿 Ω𝑓 𝑡
• 𝜔 describes the velocity of the mesh and is independent of the solution
• 𝜔 = 0 or 𝜔 = 𝑣 result in the Eulerian or Lagrangian description, respectively
• 𝜔 can be selected to minimize mesh distortion
Approximation of Solution Over the Domain
Ω𝑓
Ω𝑓𝑘
 Approximation procedure
• Subdividing the domain into K disjoint quadrilateral elements
•
Ω𝑓 =
𝐾
𝑘
𝑘=1 Ω𝑓
• We impose a piecewise polynomial approximation space defined for the Ω𝑓𝑘
• The polynomial approximation space is:
•
𝑃𝑁,𝐾 Ω𝑓 = Φ ∈ 𝐿2 Ω ; Φ|Ω𝑘 ∈ 𝑃𝑁 Ω𝑓𝑘
• The spectral element approximation space is defined as:
•
χ𝑘 = 𝐻01 Ω𝑓 ∩ 𝑃𝑁,𝐾 Ω𝑓
Imposition of Polynomial Approximation
The polynomial basis (Jacobi polynomials)takes the form:
The unknowns of the problem become:
The weak form then becomes:
Where:
•
The mass matrix
•
Derivative matrix
•
The stiffness (Laplacian) matrix
•
Fi is the vector of body forces
•
Nonlinear convective term
Time Stepping Procedure
An implicitly stiffly stable time integration scheme was applied resulting in:
Coupled with the following set of equations:
Solid Mechanics Governing Equation
The weak form of the governing equation takes the form:
It is nonlinear in nature and has to be solved by the Newton-Raphson
procedure. This procedure requires the linearization of the energy functional
A similar spatial discretization scheme is used for the solid mechanics
solution approximation. Higher order polynomials are used as basis functions
in a courser grid than used with lower order polynomials
Fluid-Structure Interaction at Interface
Kinematic compatibility conditions:
Equilibrium of tractions:
Fluid-Structure Interaction at Interface
Transfer of traction from fluid to structure:
.
𝑡𝑠 = 𝑎𝑟𝑔𝑚𝑖𝑛
Γ𝑓𝑠
𝑡𝑠 𝑥, 𝑦 − 𝑡𝑓 𝑟, 𝑠
2
.
𝑑Γ𝑓𝑠 = 𝑎𝑟𝑔𝑚𝑖𝑛
Γ𝑓𝑠
2
𝑁𝑠 𝑡𝑠 − 𝑡𝑓 𝑑Γ𝑓𝑠
Utilizing the least squares procedure:
𝑡𝑠 =
𝑀−1
𝑇
.
𝑁 𝑡𝑓 𝑑Γ𝑓𝑠
Γ𝑓𝑠 𝑠
where: M =
𝑇
.
𝑁 𝑁𝑠 𝑑Γ𝑓𝑠
Γ𝑓𝑠 𝑠
Similarly, the kinematic boundary conditions is transferred from the solid to the fluid
𝑀𝑓 𝑑𝑓 =
𝑇
.
𝑁
𝑑𝑓 Γ𝑓𝑠
Γ𝑓𝑠 𝑓
where: M𝑓 =
𝑇
.
𝑁
𝑁𝑓 𝑑Γ𝑓𝑠
Γ𝑓𝑠 𝑓
The fluid, solid and boundary are coupled in the following nonlinear form:
Application Problem
Cylinder with a flag with a flow channel:
Discretization of the Problem Domain
Solution: Velocity and Pressure Profiles
Solution: Tractions on the Upper Surface of the Flag
Author’s Speculation on Results
The discrepancies might be due to:
(a) The sharp gradients impose more difficulties to the spectral element
discretization than low order methods do.
(b) The singularity in the solution is not seen so clearly by the method
Possible modifications to improve results
Reference [4]
• Employs a mixed finite element and spectral method to discretize the domain
• Uses the low order polynomial approximation to investigate the region where
singularities are expected to take place; while the higher order spectral
discretization takes place where the solution is expected to be smooth
Test Problem
• Domain: Ω = 0 1 × [0 1]
• Test function
𝑢 𝑥, 𝑦 = 𝑥𝑦 1 − 𝑥 1 − 𝑦 ln(0.1 + 𝑥 + 𝑦)
Possible modifications to improve results
Reference [5]
• Studied the flow past a cylinder
References
1) Kollmannsberger, Stefan. ALE-type and fixed grid fluid-structure interaction
involving the p-version of the Finite Element Method. München, Techn. Univ.,
Diss., 2010.
2) Ho, Lee Wing. A Legendre spectral element method for simulation of
incompressible unsteady viscous free-surface flows. Diss. Massachusetts
Institute of Technology, 1989.
3) https://www.sharcnet.ca/Software/Ansys/fluent/14.0/help/ans_lsd/Hlp_L_alef
orm.html
4) Schneidesch, C. R., M. O. Deville, and E. H. Mund. "Domain decomposition
method coupling finite elements and preconditioned Chebyshev collocation to
solve elliptic problems." Contemporary Mathematics 157 (1994): 293-293.
5) Verkaik, A. C., et al. "Coupled Overlapping Domain method for fluid-structure
interaction."
Questions?