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SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 1
Semaine 7 Modélisation de la propagation d’ondes
guidées
Ramy Mohamed
Été 2011
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 2
A little bit of Philosophy
Physical World
Mathematical World
Engineering (Numerical World)
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 3
Plan
• Objectives of session: • What is special about guided waves, both physically and numerically.
• Identify the major Numerical Methods that is used in modeling guided waves propagation.
• Understand the major factors controlling the accuracy and validity of a numerical simulation of a guided waves propagation
• Organisation of session: • Strong Form schemes (Finite Difference & Spectral Schemes)
• Break
• Weak Form schemes ( Finite Element & Spectral Element)
• Devoir
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 4
Reminder
• At the beginning there was ODE
Given , Find a differentiable function defined for (a) (b) For all In order to have a solution, should be continuous in time and bounded in u, L > 0
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 5
Discretization
Exact Approximate
We need to achieve a prescribed accuracy with the minimum number of function evaluations.
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 6
Discretization
• Types of Errors: Discretization errors: Numerical approximation
issue. Round-off errors: Implementation issue
• Numerical Scheme Attributes • Accuracy, Stability and Convergence
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 7
FD Schemes
• Forward Operator
• Backward Operator
• Central
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 8
– Local approximation error – Exact solution – Approximate solution
Order of accuracy The numerical scheme has order of accuracy if
Accuracy
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 9
Stability of FD Schemes
• Lax-Richtmyer Stability (zero Stability) as in bounded interval [0, t] • Absolute Stability
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 10
Stability Sensitivity of the algorithm to small perturbations in input (error driven).
–Numerical Stability –Algorithmic Stability
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 11
Convergence of Explicit FD Schemes
FD Scheme is Consistent if Convergence = Stable time step + Consistency
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 12
FD Schemes
Leapfrog (2nd Order)
Crank-Nicolson (2nd Order)
Runge-Kutta
(2th Order)
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 13
Polynomial Approximation
The unknown solution Is Approximated by
Lagrange Polynomials
– An example of Algorithmic stability
– Barycentric-Remerez Algorithm is more stable
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 14
Differentiation Matrices
Interpolation Spectral
(Global Support)
FD
(local Support)
FD FD
Spectral
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 15
Spectral Schemes
• To overcome Runge Phenomenon
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1x 10
-3 equispaced points
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
equispaced points
-1 -0.5 0 0.5 1-4
-2
0
2
4x 10
-5 Chebyshev points
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
Chebyshev points
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 16
Spectral Schemes
100 101 102 103 10410-15
10-10
10-5
100
N
erro
r
Convergence of fourth-order finite differences
N-4
100 101 10210-15
10-10
10-5
100
N
erro
r
Convergence of spectral differentiation
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 17
Prelude to Guided Waves Simulation
• Types of PDEs: Elliptic : Time Independent, Equilibrium
phenomena. Parabolic : Time dependent, and diffusive Hyperbolic : Time dependent, wavelike, with a
finite speed of propagation
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 18
Prelude to Guided waves Simulation
Hyperbolic Parabolic
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 19
Wave Equation in 1D
Strong Form: find such that • Governing Equation
• Initial Conditions
• Boundary Conditions Essential Natural
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 20
FD Schemes for 1D Wave Equation
2nd Order Temporal-Spatial Scheme: Courant Friedrichs Lewy (CFL) Condition:
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 21
FEM Boundary Conditions
• Essential Boundary Condition At least one point of the boundary should have an essential BC.
• Natural Boundary Condition
Common nodes satisfy the essential BC.
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 22
FEM Static
• System of equations: strong form, difficult to solve, over complex geometry.
• Weak form: requires weaker continuity on the dependent variables.
• Weak form is often preferred for obtaining an approximated solution.
• Formulation based on a weak form leads to a set of algebraic system equations – FEM.
• FEM can be applied for practical problems with complex geometry and boundary conditions.
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 23
FEM Static
•Exact at the nodes •There exists at least one point in each element at which the derivative is exact •The derivative is 2nd order accurate at the midpoints of elements
x
F ( x )
nodes elements
Unknown function of field variable
Unknown discrete values of field variable at nodes
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 24
Preprocessing
FEM Static
• Step 1: Domain discretization (Meshing) • Step 2: Displacement interpolation (Element Order) • Step 3: Formation of FE equation in local coordinate system • Step 4: Coordinate transformation or mapping • Step 5: Assembly of FE equations • Step 6: Imposition of displacement constraints • Step 7: Solving the FE equations
Solver
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 25
FEM Static Step 1
• The solid body is divided into N elements with proper connectivity – compatibility.
• All the elements form the entire domain of the problem without any overlapping – compatibility.
• There can be different types of element with different number of nodes. (Conforming vs Non-Conforming Elements)
• The density of the mesh depends upon the accuracy requirement of the analysis.
• The mesh is usually not uniform, and a finer mesh is often used in the area where the displacement gradient is larger.
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 26
• 2D solid elements are applicable for the analysis of plane strain and plane stress problems.
• A 2D solid element can have a triangular, rectangular or quadrilateral shape with straight or curved edges.
• A 2D solid element can deform only in the plane of the 2D solid.
• At any point, there are two degrees of freedom (dofs) in the x and y directions for the displacement as well as forces.
FEM Static Step 2
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 27
FEM Static Linear Triangular Element
•Less accurate than quadrilateral elements •Used by most mesh generators for complex geometry (Unstructured Meshing) •A linear triangular element:
x, u
y, v
1 (x1, y1) (u1, v1)
2 (x2, y2) (u2, v2)
3 (x3, y3) (u3, v3)
A fsx
fsy
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 28
FEM Static Linear Rectangular Element
•Non-constant strain matrix •More accurate representation of stress and strain •Regular shape makes formulation easy
x, u
y, v
1 (x1, y1) (u1, v1)
2 (x2, y2) (u2, v2)
3 (x3, y3) (u3, v3)
2a
fsy fsx
4 (x4, y4) (u4, v4)
2b
η
ξ
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 29
FEM Static Element Distortion
• Use of distorted elements in irregular and complex geometry is common but there are some limits to the distortion.
• The distortions are measured against the basic shape of the element – Square ⇒ Quadrilateral elements – Isosceles triangle ⇒ Triangle elements – Cube ⇒ Hexahedron elements – Isosceles tetrahedron ⇒ Tetrahedron elements
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 30
FEM Static Element Distortion
• Aspect Ratio Distortion
b
a
3 Stress analysis10 Displacement analysis
ba
≤
Rule of thumb:
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 31
FEM Static Element Distortion
•Angular Distortion
skew Taper b a
b<5a
<120°
>60°
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 32
FEM Static Increasing Accuracy: h adaptation
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 33
FEM Static Increasing Accuracy: p adaptation
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 34
FEM for Guided Waves Transient Analysis
We need To march the system in time (find the solution at discrete time steps a) Accuracy b) Stability
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 35
Time Integration Transient Analysis
• The direct integration method is basically using the finite difference scheme for time marching.
• There are mainly two types of direct integration method: implicit and explicit.
• Implicit method (e.g. Newmark’s method) is more efficient for relatively slow phenomena.
• Explicit method (e.g. central differencing method) is more efficient for fast phenomena, such as impact, explosion and guided waves.
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 36
Transient Analysis
Central Difference Method
1) Explicit 2) A special initialization procedure is needed
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 37
Transient Analysis
1- Identify the frequencies contained in the loading.
2- Choose a FE mesh that can accurately represent the static, and accurately all frequencies up to about
3- Perform a Direct time integration
analysis.
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 38
Transient Analysis Amplification factor Vibration Period T
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 39
Transient Analysis
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 40
Transient Analysis
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 41
Transient Analysis
Numerical Dispersion: Higher frequency modes propagate numerically, while not representing a physical phenomena
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 42
Time Integration Transient Analysis
• Time Step (CFL Condition)
• For 1% Numerical Dispersion 20 Points per minimum Wavelength
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 43
Spectral Element
Mapping (Superparametric)
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 44
Spectral Element
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 45
Spectral Element
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 46
Spectral Element
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 47
Numerical Dispersion
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 48
CPU time
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 49
Spectral Element
S0 interaction with Notch
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 50
Spectral Element
A0 interaction with notch
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 51
SEM vs FEM
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 52
Spectral Element Time stability FEM mesh (quartic Quad)
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 53
Spectral Element Time stability SEM mesh (Quad 3X6)
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 54
Thanks
Questions
SURVEILLANCE DES STRUCTURES – GMC 724 Modélisation ondes guidées 55
Types of PDEs
2
0( 0
0 - 4ac)
Ellipticb Parabolic
Hyperbolic
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