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X-FEM for FSI Interfaces
Andreas Kolke
Institute for Structural AnalysisTechnische Universitat Braunschweig
03 May 2006
Advanced Computational Methods
for Fluid-Structure Interaction
Aim of the Talk
Aim of the TalkConceptional Use ofX-FEM for FSIProblems
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
2 / 40
Conceptional Use of X-FEM for FSI Problems
Aim of the TalkConceptional Use ofX-FEM for FSIProblems
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
3 / 40
Physics:
Modeling coupled problems Fluid-structure interaction: special solutions? A priori knowledge of solution characteristics in FSI?
Numerics:
Strategies of discretization X-FEM X-FEM for FSI (space-time finite element method)
Interaction of Fluid and Structure
Aim of the Talk
Interaction of Fluidand StructureOscillators &Mechanisms ofExcitation
Examples
Coupled Problem
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
4 / 40
Oscillators & Mechanisms of Excitation
Aim of the Talk
Interaction of Fluidand StructureOscillators &Mechanisms ofExcitation
Examples
Coupled Problem
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
5 / 40
Oscillators:
Body oscillatorsstructures of discrete/distributed mass/elasticity
Fluid oscillatorsfluids of discrete/distributed mass/compressibility
Mechanisms of Excitation:
Extraneously induced excitationvariations in the flow field (turbulence, noise)
Instability-induced excitationvariations due to flow-intrinsic instability(vortex shedding, shear layers, sensitive interfaces)
Movement-induced excitationvariations due to changes of geometry(flutter, galloping, slender structures in axial flow, ...)
Examples
Aim of the Talk
Interaction of Fluidand StructureOscillators &Mechanisms ofExcitation
Examples
Coupled Problem
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
6 / 40
Coupled Problem
Aim of the Talk
Interaction of Fluidand StructureOscillators &Mechanisms ofExcitation
Examples
Coupled Problem
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
7 / 40
Multiple bodies in interaction via boundary or domain.
s
nΩ2Ω1
Γ2
Γ1
Σ
Figure 1: boundary-coupled two-field system.
Model equations for fluid & structure,Boundary and initial conditions,Model equations for coupling at common interface.
Characteristic Solutions
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutionsSpecial SolutionProperties with FSI
General Approach
Thin Structures
Tips and Edges
Wall Effects
Bluff Bodies
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
8 / 40
Special Solution Properties with FSI
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutionsSpecial SolutionProperties with FSI
General Approach
Thin Structures
Tips and Edges
Wall Effects
Bluff Bodies
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
9 / 40
Fixed structures may induce flow instabilities Motion of structures can change the flow pattern Characteristics of solution dependent on model
(but their type is often known a priori !)
fluid: viscous/inviscid,compressible/incompressible structure: thin-walled, tips, edges coupling: no-slip, slip, damping, friction
Solutions of special type are usually non-smooth:
strong or weak discontinuous singular logarithmic ...
General Approach
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutionsSpecial SolutionProperties with FSI
General Approach
Thin Structures
Tips and Edges
Wall Effects
Bluff Bodies
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
10 / 40
Demand: develop & apply simple models to use as afirst approximation to a complex FSI process
Action: accept resulting type of expectable solution
Advantage: avoid time-consuming resolution of fullmodel’s length and time scales
Disadvantage: need for sophisticated solutiontechnique to capture solution properties
Thin Structures
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutionsSpecial SolutionProperties with FSI
General Approach
Thin Structures
Tips and Edges
Wall Effects
Bluff Bodies
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
11 / 40
Discontinuous flow state variables from embedding aLagrangian structure of zero thickness into an Eulerianfluid field
e.g. for velocity-pressure N-S-eq. & no-slip interfaceconditions: strongly discontinuous pressure solutionand weakly (gradient) discontinuous velocity solutionalong the moving fluid-structure interface
slip conditions: tangential velocity component mayexhibit a jump discontinuity
Tips and Edges
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutionsSpecial SolutionProperties with FSI
General Approach
Thin Structures
Tips and Edges
Wall Effects
Bluff Bodies
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
12 / 40
Geometry of boundaries/structure may cause singularvalued quantities in physical fields
Sensitivity of coupled systems (very light structures,...)to singularities
Existence of singularities identifiable by inspection ofgeometry
Particularly, if geometry is modified (FSI: structuraldeformation or motion) singular fluid solutions shouldbe captured accurately by the numerical solution.
Wall Effects
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutionsSpecial SolutionProperties with FSI
General Approach
Thin Structures
Tips and Edges
Wall Effects
Bluff Bodies
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
13 / 40
Representation of (turbulent) boundary layer Wall functions
Reduce computational costs by using the a prioriknowledge of logarithmic type of the solution
Location of the boundary layer is defined by theinterface position
Bluff Bodies
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutionsSpecial SolutionProperties with FSI
General Approach
Thin Structures
Tips and Edges
Wall Effects
Bluff Bodies
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
14 / 40
Representation of impinging shear layers, sheddingvortices or interface instabilities caused by bluff bodiesor fixed/moving boundaries
location of special solutions in the vicinity of interfacesare easiliy allocable
Approximation of Special Solutions
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
DomainDiscretization
Adaptivity
Choosing the Basis
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
15 / 40
Domain Discretization
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
DomainDiscretization
Adaptivity
Choosing the Basis
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
16 / 40
Geometry approximation of fluid and solid configurationinvolved in the multi-field problem:
Explicit description methods(boundary-fitted or interface-fitted)→ fixed connectivity, mesh-moving/remeshing
Implicit description methods(indicator/marker functions)→ changing connectivity, no mesh-moving/remeshing
Adaptivity
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
DomainDiscretization
Adaptivity
Choosing the Basis
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
17 / 40
Mesh adaptivity (assess the quality of the results):
Moving nodes (r-adaptivity) Refining (and coarsening) elements (h-adaptivity) Changing order of base functions (p-adaptivity) Combinations of the above (e.g. hp-adaptivity)
www.diffpack.com
Choosing the Basis
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
DomainDiscretization
Adaptivity
Choosing the Basis
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
18 / 40
Choosing the basis of approximation:
Higher degree piecewise polynomial basis functions(spectral element methods)
Rational basis functions Extrinsic enriched approximation using the
Partition of Unity Method (PUM) → PU-FEM, X-FEM
Appropriate modification of the ansatz to the approximation so that
the function space of the modified approximation is made able to
reflect the essential properties of the solution to the model equations.
X-FEM (I): Introduction
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
Partition of Unity I
Partition of Unity II
Extrinsic Enrichmentof FE Approximation
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
19 / 40
Partition of Unity I
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
Partition of Unity I
Partition of Unity II
Extrinsic Enrichmentof FE Approximation
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
20 / 40
A set of functions Nk is called a partition of unity (PU) if
X
k
Nk(x)p(xk) = p(x) ∀ x ∈ Ω, (1)
where p is a polynomial basis of order n.
Within a weighted residual formulation these functions are able torepresent any analytical solution of polynomial type up to order nexactly.
Partition of unity methods 1 2 enrich the approximation space by anadditional basis g and choose an ansatz of the form
uhPU (x) =
X
k
Nk(x)g · uk (2)
with modified properties of approximation.
1I. Babuska and J. M. Melenk. The Partition of Unity Method. IJNME, 40:727–758, 1997.
2J. M. Melenk and I. Babuska. The Partition of Unity Finite Element Method: Basic Theory and
Applications. CMAME, 39:289–314, 1996.
Partition of Unity II
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
Partition of Unity I
Partition of Unity II
Extrinsic Enrichmentof FE Approximation
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
21 / 40
The basis g may incorporate
Lagrange or Taylor polynomials, Discontinuous functions (Heaviside, Dirac delta, absolute value), Singular functions, Transcendental functions (trigonometric, logarithmic)
or other appropriate expressions.
Any available a priori known information on thecharacteristic properties of the expected solution tothe modeled problem should be taken into accountwhile composing the enriched basis g.
Extrinsic Enrichment of FE Approximation
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
Partition of Unity I
Partition of Unity II
Extrinsic Enrichmentof FE Approximation
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
22 / 40
The partition of unity ansatz can be expressed by an extrinsicenrichment using
g =
»
1ψ
–
and uk =
»
uk
u∗k
–
(3)
to extend the standard approximation (of ansatz functions Nk) by theproducts Nkψ.
The enrichment function ψ modifies the standard approximation andcreates an extended approximation
uhext(x) =
X
k∈N std
Nk(x) uk +X
j∈N ext
Nj(x)ψ(x) u∗j (4)
with additional coefficients u∗j .
X-FEM (II): Evolving Solutions
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
EvolvingNon-smoothSolutionsSpace-Time FiniteElement MethodExtrinsic Enrichmentof Space-Time FEM
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
23 / 40
Evolving Non-smooth Solutions
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
EvolvingNon-smoothSolutionsSpace-Time FiniteElement MethodExtrinsic Enrichmentof Space-Time FEM
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
24 / 40
A dynamical problem may imply propagating and evolving solutions ofspecial type.
y
x
t
Q
y
Qn
n−1
n+1
x
t
A priori knowledge on instationary solution behavior/characteristics →proper enrichment of a space-time approximation with finite elements.
Local PU: use of implicit methods for localizing the enrichment, e.g.
one or multiple level set functions 3.
3S. Osher and J. A. Sethian. Fronts Propagating with Curvature-Dependent Speed: Algorithms
Based on Hamilton-Jacobi Formulations. JCP, 79:12–49, 1988.
Space-Time Finite Element Method
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
EvolvingNon-smoothSolutionsSpace-Time FiniteElement MethodExtrinsic Enrichmentof Space-Time FEM
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
25 / 40
Methodically uniform discretization of space and time applyingthe finite element approach (Oden, Argyris and Fried)
y
x
t
Q
y
Qn
n−1
n+1
x
t
Straightforward applicability of X-FEM technology to describeevolving non-smooth solutions within the space-time domain
Enriched space-time finite elements enables to capturestrong/gradient discontinuities, singularities, boundary layerspropagating through the domain
Approximation in space-time with the unknown coefficients uk:
uhstd(x, t) =
X
k∈N std
Nk(x, t) uk (5)
Extrinsic Enrichment of Space-Time FEM
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
EvolvingNon-smoothSolutionsSpace-Time FiniteElement MethodExtrinsic Enrichmentof Space-Time FEM
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
26 / 40
Combination of level set method, local partition of unityenrichment & space-time finite element discretization for evolvingsolution features – discontinuities at thin-walled structures 4 5
Extended approximation in space-time with coefficients uk andadditional unknowns u∗
j of the enrichment:
uhext(x, t) =
X
k∈N std
Nk(x, t) uk +X
j∈N ext
Nj(x, t)ψj(x, t) u∗j (6)
The nodally defined enrichment function
ψj(x, t) = ψ(x, t) − ψ(xj , tj) (7)
incorporates placement of local enrichment within the space-time
finite element and ensures Kronecker-δ property of the approximation.
4A. Legay and A. Kolke. A locally enriched space-time finite element method for fluid-structure
interaction – Part I: Prescribed structural motion. in preparation5A. Kolke and A. Legay. A locally enriched space-time finite element method for fluid-structure
interaction – Part II: Thin flexible structures. in preparation
X-FEM (III): Enriching the Flow
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
Thin Structures:Discontinuities
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
27 / 40
Thin Structures: Discontinuities
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
Thin Structures:Discontinuities
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
28 / 40
Strong and weak discontinuities in a velocity-pressure fluidformulation realized by jump-type enrichment function ψj(x, t)
Locally enriched velocity/pressure approximation:
vhext(x, t) =
X
k∈N std
Nk(x, t) vk +X
j∈N ext
Nj(x, t)ψj(x, t) v∗j
phext(x, t) =
X
k∈N std
Nk(x, t) pk +X
j∈N ext
Nj(x, t)ψj(x, t) p∗j (8)
with additional unknowns v∗j and p∗j .
Continuity of the velocity field (no-slip) at the fluid-structureinterface to be ensured by an appropriate method.
Enrichment function determined by evaluation of the level setfunction φ
ψj(x, t) =1
2(1 − sign φ(x, t) · sign φ(xj , tj)) (9)
at the nodes j and at the point within the domain.
X-FEM (IV): Coupling Conditions
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
Penalty Methods
Lagrange MultiplierMethodsDistributed LMMethod
MappingConfigurations
X-FEM for FSI:Applications
29 / 40
Penalty Methods
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
Penalty Methods
Lagrange MultiplierMethodsDistributed LMMethod
MappingConfigurations
X-FEM for FSI:Applications
30 / 40
Add a penalty term of constraint to the weak form
α
∫
R
δw(u+− u
−)dR (10)
Factor α >> 1 Constraint accuracy increases with large numbers of α
Advantage: size of system of equations is constant Disadvantage: influences condition number of resulting
system of equations, ill-conditioning
Nitsche’s method(consistent improvement of the penalty method):several problem dependent penalty terms with α-valuesconsiderably smaller
Lagrange Multiplier Methods
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
Penalty Methods
Lagrange MultiplierMethodsDistributed LMMethod
MappingConfigurations
X-FEM for FSI:Applications
31 / 40
Add a Lagrange multiplier formulation of constraint tothe weak form∫
R
δλ(u+− u
−)dR −
∫
R
δ(u+− u
−)λdR (11)
Advantage: General and accurate approach Disadvantage: LM need to be solved additionally to
discrete field variables, set of interpolation functionsfor LM required
Discretization of λ for ’evolving’ constraints?(FSI problems on fixed meshes, ...)
Distributed LM Method
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
Penalty Methods
Lagrange MultiplierMethodsDistributed LMMethod
MappingConfigurations
X-FEM for FSI:Applications
32 / 40
Since the fluid-structure interface is moving and deforming thediscrete description of interface tractions in an explicit manner isnot easy to realize.
Apply an implicit technique (level set approach) as it is realizedfor interface representation.
u1
u2
φ = 0
tt
LM λ(x, t) represented by zero level set φ(x, t) = 0 ofhigher-dimensional function λ(x, t)
λ(x, t) = λ(x, t) δD(φ(x, t)) , (12)
where δD is the Dirac delta function.
Implicit formulation of the Lagrangian multiplier!
Mapping Configurations
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
Level SetRepresentation ofThin-walledStructures
X-FEM for FSI:Applications
33 / 40
Level Set Representation of Thin-walled Structures
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
Level SetRepresentation ofThin-walledStructures
X-FEM for FSI:Applications
34 / 40
The structure is projected to the flow using a level set function.
approximated zero level-set
discretized thin structure tn tn+1
Updated level set function φ derived from the current configuration ofthe thin-walled structure (curve/surface-shaped interface):
φ(x, t) = ±min(‖x(t) − xR(t)‖2) (13)
Mapping: approximation error depending on mesh resolution. 6
6A. Legay, J. Chessa, and T. Belytschko. An Eulerian-Lagrangian Method for Fluid-Structure
Interaction Based on Level Sets. CMAME, 195(17-18):2070–2087, 2006.
X-FEM for FSI: Applications
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
Piston I
Piston II
Piston III
Supported Beam
Rotating Beam
35 / 40
Piston I
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
Piston I
Piston II
Piston III
Supported Beam
Rotating Beam
36 / 40
L
H
b
k
k
ρF
µF
ρS
vSt p
H = 1.0, L = 10.0, b = 2.5
p = 0.0, ρF = 1.0, µF = 1.0
Piston 1: k = 0.0, t = 0.0, vS = 10 · t
Piston 2: k = 0.2, t = 1.0, ρS = 10−8
Piston II
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
Piston I
Piston II
Piston III
Supported Beam
Rotating Beam
37 / 40
Piston 1: Structure with prescribed constant acceleration
The pressure solution is linear, but discontinuous at the FSI interface.
Piston III
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
Piston I
Piston II
Piston III
Supported Beam
Rotating Beam
38 / 40
Piston 2: Spring supported structure, constant inflow traction
Supported Beam
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
Piston I
Piston II
Piston III
Supported Beam
Rotating Beam
39 / 40movie
Rotating Beam
Aim of the Talk
Interaction of Fluidand Structure
CharacteristicSolutions
Approximation ofSpecial Solutions
X-FEM (I):Introduction
X-FEM (II):Evolving Solutions
X-FEM (III):Enriching the Flow
X-FEM (IV):Coupling Conditions
MappingConfigurations
X-FEM for FSI:Applications
Piston I
Piston II
Piston III
Supported Beam
Rotating Beam
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movie: pressure field movie: level set function
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